Statistics Toolbox

Fitting the Hougen-Watson Model

The Statistics Toolbox provides the function nlinfit for finding parameter estimates in nonlinear modeling. nlinfit returns the least squares parameter estimates. That is, it finds the parameters that minimize the sum of the squared differences between the observed responses and their fitted values. It uses the Gauss-Newton algorithm with Levenberg-Marquardt modifications for global convergence.

nlinfit requires the input data, the responses, and an initial guess of the unknown parameters. You must also supply the name of a function that takes the input data and the current parameter estimate and returns the predicted responses. In MATLAB terminology, nlinfit is called a "function" function.

Here is the hougen function.

function yhat = hougen(beta,x)
%HOUGEN Hougen-Watson model for reaction kinetics.
%   YHAT = HOUGEN(BETA,X) gives the predicted values of the
%   reaction rate, YHAT, as a function of the vector of 
%   parameters, BETA, and the matrix of data, X.
%   BETA must have five elements and X must have three
%   columns.
%
%   The model form is:
%   y = (b1*x2 - x3/b5)./(1+b2*x1+b3*x2+b4*x3)
b1 = beta(1);
b2 = beta(2);
b3 = beta(3);
b4 = beta(4);
b5 = beta(5);
x1 = x(:,1);
x2 = x(:,2);
x3 = x(:,3);
yhat = (b1*x2 - x3/b5)./(1+b2*x1+b3*x2+b4*x3);

To fit the reaction data, call the function nlinfit.

load reaction
betahat = nlinfit(reactants,rate,'hougen',beta)
betahat =
    1.2526
    0.0628
    0.0400
    0.1124
    1.1914

nlinfit has two optional outputs. They are the residuals and Jacobian matrix at the solution. The residuals are the differences between the observed and fitted responses. The Jacobian matrix is the direct analog of the matrix X in the standard linear regression model.

These outputs are useful for obtaining confidence intervals on the parameter estimates and predicted responses.

Example: Nonlinear Modeling

Confidence Intervals on the Parameter Estimates